Higher order theory for analytic saddle point approximations to the PP and PS reflected arrivals at a solid/solid interface
P. F. Daley
The high frequency solution to the problem of a PP and PS reflected wave at a plane interface between two isotropic homogeneous halfspaces at near grazing incidence is developed. In reflection seismology sub-critical reflections are often all that is required in data processing. A distorted (compared to the zero order geometrical optics approximation to the arrival) wavelet is seen on the traces in the area of grazing incidence. For a proper velocity distribution, there may an area in the vicinity of the critical point which also requires special treatment. This results in the zero order geometrical optics approximation being supplemented by a term involving Parabolic Cylinder Function of the order. Grazing incidence may occur in cross hole tomography or for a point source in a thin surface layer with a large velocity contrast at the interface with the subsequent layer.